Olympiad problems

2007 Sharygin Geometry Olympiad, 4

Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A' $ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

1998 Putnam, 6

Tags: geometry

Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

1984 All Soviet Union Mathematical Olympiad, 389

Tags: sequence , limit , algebra , recurrence relation

Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.

2015 China Team Selection Test, 1

Tags: perpendicular bisector , angle bisector , geometry

$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.

2024 Dutch IMO TST, 4

Tags: combinatorics

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

2003 Alexandru Myller, 4

Tags: function , calculus , integration

[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ [i]Mihai Piticari[/i]

2015 Purple Comet Problems, 6

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There are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99. Find 10a + b.

2015 HMIC, 4

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Prove that there exists a positive integer $N$ such that for any positive integer $n \ge N$, there are at least $2015$ non-empty subsets $S$ of $\{ n^2 + 1, n^2 + 2, \dots, n^2 + 3n \}$ with the property that the product of the elements of $S$ is a perfect square.

2010 Contests, 3

Tags: power of a point , radical axis , geometry

Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

2019 AIME Problems, 14

Tags: number theory

Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.

2016 KOSOVO TST, 3

Tags: algebra , quadratic

Equations $x^2+ax+b=0$ and $x^2+px+q=0$ have a common root.Find quadratic equation roots of which are two other roots.

2018 Dutch Mathematical Olympiad, 2

Tags: combinatorics , coloring

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$, then $x \cdot y$ is red.

1983 IMO Longlists, 40

Tags: ratio , geometry , 3d geometry , tetrahedron , sphere , function , congruent triangles

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

2019 China Team Selection Test, 3

Tags: algebra

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

2007 Federal Competition For Advanced Students, Part 2, 3

Tags: rhombus , geometry

Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha \equal{} \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.

2003 Gheorghe Vranceanu, 3

Tags: modular arithmetic , number theory

Show that $ n\equiv 0\pmod 9 $ if $ 2^n\equiv -1\pmod n, $ where $ n $ is a natural number greater than $ 3. $

2006 Regional Competition For Advanced Students, 4

Tags: algebra

Let $ <h_n>$ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $ h_n$ is the harmonic mean of its two neighbours $ h_{n\minus{}1}$ and $ h_{n\plus{}1}$ : $ h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}$) Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers.

2006 Estonia National Olympiad, 5

Tags: combinatorics

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.

2014 ELMO Shortlist, 6

Tags: inequalities

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2022 Estonia Team Selection Test, 4

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 11.7

Tags: number theory , coprime

Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.

2006 District Olympiad, 2

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For a positive integer $n$ we denote by $u(n)$ the largest prime number less than or equal to $n$, and with $v(n)$ the smallest prime number larger than $n$. Prove that \[ \frac 1 {u(2)v(2)} + \frac 1{u(3)v(3)} + \cdots + \frac 1{ u(2010)v(2010)} = \frac 12 - \frac 1{2011}. \]

2014 Singapore MO Open, 2

Tags: function , algebra

Find all functions from the reals to the reals satisfying \[f(xf(y) + x) = xy + f(x)\]

2019 China Northern MO, 3

Tags: algebra

$n(n\geq2)$ is a given intenger, and $a_1,a_2,...,a_n$ are real numbers. For any $i=1,2,\cdots ,n$, $$a_i\neq -1,a_{i+2}=\frac{a_i^2+a_i}{a_{i+1}+1}.$$ Prove: $a_1=a_2=\cdots=a_n$. (Note: $a_{n+1}=a_1,a_{n+2}=a_2$.)

2004 India IMO Training Camp, 2

Tags: polynomial , algebra

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$